The homology of special linear groups over polynomial rings, by Kevin P. Knudson

In this paper we study the homology of the group SL_n(F[t,t^{-1}]) where F is an infinite field. Our approach is to study the action of the group on the Bruhat-Tits building associated to an n-dimensional vector space over the field of Laurent series over F. A fundamental domain for this action is an (n-1)-simplex C and the structure of the stabilizers of the various faces of C is easily described. The main result of the paper is the following. For each p, the stabilizers of the p-simplices of C break up into isomorphism classes in such a way that in each class, there is a group G which fits into a split short exact sequence

1 --> K --> G --> P --> 1
where G is a subgroup of SL_n(F[t]), P is a parabolic subgroup of SL_n(F), and K consists of those matrices which are congruent to the identity modulo t. The map G --> P is given by evaluation at t=0. Then we have the following result.

Theorem. The inclusion P --> G induces an isomorphism on integral homology.

In the case p=0 (the vertex stabilizers), the group G is precisely SL_n(F[t]) and the group P is SL_n(F). In this case the theorem reduces to an unstable analogue of homotopy invariance in algebraic K-theory; namely, the integral homology of SL_n(F[t]) is the same as that of SL_n(F).

The theorem completes the computation of the E^1-term of the spectral sequence. The differential is difficult to calculate in general, but we are able to compute some special cases. In particular, we show that if F is an infinite field, then for n at least 3, the group H_2(SL_n(F[t,t^{-1}],Z) equals the direct sum of H_2(SL_n(F),Z) and the group of units of F.

This paper has been submitted for publication in the Annales Scientifiques de l'Ecole Normale Superieure.


Kevin P. Knudson <knudson@math.duke.edu>